In this paper we study the power-like confining potentials r α . The region of allowed α's is identified, the mass-gap equation is constructed for an arbitrary α and solved for several values of the latter, and the vacuum energy and the chiral condensate are calculated. The question of replica solutions to the mass-gap equation for such potentials is addressed and it is demonstrated that the number of replicas is infinite for any α, as a consequence of the peculiar behaviour of the quark self-energy in the infrared domain.